DAGnabbit! Ensuring Consistency between Noise and Detection in Hierarchical Bayesian Inference

TL;DR

This paper highlights the importance of correctly modeling detection processes in hierarchical Bayesian inference for astrophysics, showing that ignoring the physical detection process can lead to biases in population inferences.

Contribution

It demonstrates that common approximations in hierarchical Bayesian models are incompatible with physical detection processes, leading to biases, and recommends simulating detected data for accurate inference.

Findings

Incorrect modeling causes biases in astrophysical population estimates.

Simulating detected data improves inference accuracy.

Biases can affect conclusions about gravity theories and merger rates.

Abstract

Hierarchical Bayesian inference can simultaneously account for both measurement uncertainty and selection effects within astronomical catalogs. In particular, the hierarchy imposed encodes beliefs about the interdependence of the physical processes that generate the observed data. We show that several proposed approximations within the literature actually correspond to inferences that are incompatible with any physical detection process, which can be described by a directed acyclic graph (DAG). This generically leads to biases and is associated with the assumption that detectability is independent of the observed data given the true source parameters. We show several examples of how this error can affect astrophysical inferences based on catalogs of coalescing binaries observed through gravitational waves, including misestimating the redshift evolution of the merger rate as well as incorrectly inferring that General Relativity is the correct theory of gravity when it is not. In general, one cannot directly fit for the ``detected distribution'' and ``divide out'' the selection effects in post-processing. Similarly, when comparing theoretical predictions to observations, it is better to simulate detected data (including both measurement noise and selection effects) rather than comparing estimates of the detected distributions of event parameters (which include only selection effects). While the biases introduced by model misspecification from incorrect assumptions may be smaller than statistical uncertainty for moderate catalog sizes (O(100) events), they will nevertheless pose a significant barrier to precision measurements of astrophysical populations.